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If $\underset{0}{\overset{\pi }{\int }}\frac{x\sin x}{1+{\sin }^{2}x}dx=\frac{\pi }{a}\ln \left(\frac{b+1}{c+1}\right);a,b,c\in R.$ Then which of the following is/are correct ?

Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is

How many solutions do the given equations have?
2+ 3= 8,4+ 6= 7.

The value of $a$ for which the curves $y^2=4x$ and $x^2+y^2-2ax=0$ intersect exactly at one point is

The period of $f\left(x\right)=\sin \left(\frac{\pi x}{n-1}\right)+\cos \left(\frac{\pi x}{n}\right),n\in Z,n>2$ is

Three concentric circles of which the biggest is $x^2+y^2=1$, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is

If two variates X and Y are connected by the relation $Y=\frac{aX+b}{c}$ , where a,b, c are constants such that ac< 0, then

For universal set $U$, and sets $A,B$ which are subsets of $U$, the following information is given

$n\left(U\right)=47,$

$n\left(A\right)=18,$

$n\left(B\right)=11,n(A\cap B)=10$



Then, the number of elements that are neither in $A$ nor $B$ is__

If $I_{m,n}=\int x^m(\ln x{)}^{n}dx,$ then $I_{m,n}-\frac{x^{m+1}}{m+1}(\ln x{)}^n=$

(where $m,n\in N;m,n\ge 1$)

For the complex number $z=\sqrt{3}i-i-1-\sqrt{3},$ the correct option(s) is/are

Find the set of values of x for which the expansion of ${(9+5x)}^{\frac{-1}{2}}$ is valid in ascending powers of x.

If $y-mx+c=0,$ $m,c\in R$ is a tangent at point $A$ to the curve $y^2=2x^2-2x,$ meeting the curve again at point $B,$ then which of the following conditions holds TRUE?

1. 0.64

If the roots of $x^2$ - (a - 3)x + a = 0 are such that at least one of the root is greater than 2, then find the range of $a$.

The multiplicative inverse of $\frac{1+5i}{26}$ is

A random variable $X$ has probability distribution



$\begin{array}{ccccccccc}X& 1& 2& 3& 4& 5& 6& 7& 8\\ P\left(X\right)& 0.13& 0.22& 0.12& 0.21& 0.13& 0.08& 0.06& 0.05\end{array}$



If events are $E=\left\{x\text{is an odd number}\right\},$$F=\left\{x\text{is divisible by}3\right\}$ and $G=\left\{x\text{is less than}7\right\}$, then the value of $P(E\cup (F\cap G\left)\right)$ is

If the number of elements in A,A-B and B-A are 8, 5, 3 respectively, then the number of elements in $A\cup BandA\cap B$ are respectively

The range of values of $x$ which satisfies the inequation ${\log }_{1/6}(x^2-3x+2)+1<0$ is

Let $A=\{x:x^2-9\le 0,x\in Z\}$ and $B=\{x:|x-2|<3,x\in Z\}.$ Then the number of elements in $A\Delta B$ is

If $f\left(x\right)$ is continuous and differentiable over $[-2,5]$ and $-4\le f^{\prime }\left(x\right)\le 3$ for all $x$ in $(-2,5),$ then the greatest possible value of $f\left(5\right)-f(-2)$ is

If $\underset{\pi /3}{\overset{x}{\int }}\sqrt{3-{\sin }^2}tdt+\underset{0}{\overset{y}{\int }}\cos tdt=0,$ then $\frac{dy}{dx}=$

The imaginary part of ${(3+2\sqrt{-54})}^{1/2}-{(3-2\sqrt{-54})}^{1/2}$ can be:

Calculate the mean and the median for the following continuous frequency distributions.

$\begin{array}{cccccccc}Class& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70\\ f_i& 6& 8& 20& 25& 15& 10& 4\end{array}$

If $sinx+si{n}^{2}x=1$, then write the value of $co{s}^{8}x+2co{s}^{6}x+co{s}^{4}x.$

The value of $\int \sqrt{\frac{1-\sin 2x}{1-\cos 2x}}dx$ is

If $\alpha$ is a root of equation $(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x$, $\beta$ is a root of equation $3{\cos }^{2}x-10\cos x+3=0$ and $\gamma$ is a root of equation $1-\sin 2x=\cos x-\sin x;0\le \alpha ,\beta ,\gamma \le \frac{\pi }{2},$ then $\sin \alpha +\sin \beta +\sin \gamma$ can be equal to

If number of terms in the expansion of $(x-2y+3z{)}^n$ are 45, then n =

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

If ${\sum }_{i=1}^{18}(x_i-8)=9and{\sum }_{i=1}^{18}(x_i-8{)}^2=45$ then the standard deviation of $x_1,x_2,\dots ..x_{18}$ is

Product of two odd functions is

The locus of the image of the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with respect to any of the tangents to the ellipse is

The line segment joining the points $(1,2)$ and $(-2,1)$ is divided by the line $3x+4y=7$ in the ratio

If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z–(3+3i)|$ is